Ms. Abedin’s work on the Sept. 20, 2012, event at the glamorous Essex House in New York City, helped entertain potential Teneo clients, wowing them with access to three former world leaders on a single stage.

It was one of the specific projects she worked on with Teneo during a seven-month period in which she earned a $15,000-a-month consulting fee from the firm while simultaneously receiving pay as a “special government employee” advising Mrs. Clinton at the State Department, according to interviews and documents.

Ms. Abedin, the wife of former Rep. Anthony D. Weiner of New York, worked as a full-time government employee and deputy chief of staff to Secretary of State Clinton from 2009 through mid-2012. She then moved to New York and transitioned to a part-time employee at State after giving birth to her first child and seeing her husband resign his congressional seat because of a sexting scandal.

In all, Ms. Abedin was paid about $105,000 over seven months to advise Teneo in New York from summer 2012 to early 2013, according to a person familiar with the arrangement. During the same time, she collected $126,239.80 in pay as a special government employee at the State Department, according to internal department records identifying her pay and leave that were obtained by The Times.

The specific nature of Ms. Abedin’s duties at Teneo has been shrouded in mystery and has become the subject of State Department and congressional inquiries looking into whether the work arrangements were proper or created any conflicts of interest.

Several sources agreed to describe Ms. Abedin’s work for Teneo on the New York event and her subsequent work at the State Department for a Teneo-connected event in Ireland four months later in December, solely on the condition of anonymity because of the ongoing investigations.

“Huma’s role for the New York event was on behalf of Teneo, while her work for the Ireland event was clearly on behalf of the State Department. But the firm, the Clintons and Huma were intertwined, and that makes it hard to separate when looking back in hindsight,” one person directly familiar with the circumstances said.

Added another source: “Teneo didn’t need Huma to get Bill Clinton or the other luminaries to the event, but she was hired to help make the event successful in terms of who attended and what they experienced.”

Teneo officials did not respond to calls seeking comment. Teneo, based in New York, employed Mr. Clinton as a consultant for about a year when it started, along with several other people from the Clinton circle. Mr. Clinton, who commands up to $700,000 per speech, did not charge Teneo for his appearance at the New York event, according to financial disclosure statements released by his wife.

Ms. Abedin’s attorney, Miguel Rodriguez, declined to comment publicly for this article. But he has steadfastly insisted that Ms. Abedin did nothing wrong in working for Teneo, the State Department and the Clinton Foundation at the same time, noting that government officials formally approved the arrangement.

Ethics questions

Ethics analysts see it differently. They say the State Department put Ms. Abedin in a position to collect money from an outside firm that had interests and political connections inside the department where she still worked, creating at least the appearance of a conflict of interest.

“The guidelines for becoming a special government employee make it quite clear it’s a unique position offered to someone as long as there is no conflict of interest,” said Craig Holman, the government affairs lobbyist for the Public Citizen ethics watchdog. “With Huma, she was in such a powerful position, conflicts were inevitable.

“Huma never should have been provided special government employee status. She should’ve either remained a full-time employee at State or else become a full-time employee in the private sector,” he said. “To merge the two does raise a serious conflict of interest.”

Added Scott Amey, general counsel for the Project on Government Oversight: “It would have been more appropriate to avoid any potential conflicts and not work inside the State Department and at the same time work for an entity that had matters before the agency. Even if this is legal or received a waiver, it certainly doesn’t mean that it should have been allowed.”

Internal State Department emails show the line dividing official government business, Teneo’s private business and the Clintons’ own financial interests at times became blurred as Ms. Abedin and other top State officials dealt with requests that had little to do with their government jobs.

The New York event, which Ms. Abedin helped support for Teneo, was held just nine days after the horrific terrorist attack on a U.S. compound in Benghazi, Libya, that killed U.S. Ambassador J. Christopher Stevens and three other Americans.

In fact, Mrs. Clinton was on Capitol Hill briefing lawmakers in private about the tragedy just a few short hours before her husband entertained the Teneo event at the Essex House overlooking South Central Park, her official schedule shows. Ms. Abedin joined Mr. Clinton at the event while Mrs. Clinton remained in Washington, according to interviews.

‘Urgent’ matter

The next morning, State Department officials were scrambling to address a potential crisis. But it was far from Libya and had nothing to do with federal business.

In their official government emails, top aides to Mrs. Clinton, including Chief of Staff Cheryl Mills and National Security Adviser Jake Sullivan, were discussing a private fundraising charity that would allow the Clintons to raise money on an upcoming trip to Ireland that had deep connections to Teneo’s clients and one of the firm’s founders, Declan Kelly.

Doug Band, a founding partner at Teneo and a longtime confidant to Mr. Clinton, was also included on the State Department email chain, which clearly had nothing to do with official business.

“Stella O’Leary called to say she saw HRC this week and that HRC ‘firmly instructed’ her to urgently form a [501(c)(3) nonprofit organization] called Friends of the Clinton Centre,” said an email from Amitabh Desai, the Clinton Foundation’s foreign policy director, dated Sept. 21, 2012.

The email was addressed to Ms. Mills, Mr. Sullivan, Mr. Band and Ms. Abedin, among others. “I also asked if the new org could be flexible so that any funding raised could be used in whatever manner WJC [initials of Mr. Clinton] and HRC [Mrs. Clinton’s initials] wish in Ireland and Northern Ireland, and not restricted to support only the current iteration of the Clinton Centre in Enniskillen.”

Ms. O’Leary is an active Democratic donor who founded the Irish American Democrats to support Mr. Clinton’s re-election in 1996 and has since been a top Clinton donor.

Ms. Abedin weighed in on the idea of setting up the fund, suggesting that Mr. Sullivan had witnessed the conversation. “Jake was in the meeting so he can verify,” Ms. Abedin wrote the following day, adding the qualifier, “but HRC had said she made no commitments to her.”

Irish program

The tax-exempt fundraising arm was indeed established in 2013 and raised about $55,000 for an international summer school program to bring children from the Balkans and other conflict zones to Northern Ireland, Ms. O’Leary told CNN last month.

Eventually, Ms. Abedin would play a critical role in setting up the Ireland trip, a sort of farewell tour for Mrs. Clinton in a country where she and her husband were beloved. Teneo’s involvement in the trip was extensive.

The host, the World Ireland Fund, was connected to Teneo executives, and a Teneo representative was listed as a contact for the event honoring Mrs. Clinton. During her speech at the event, Mrs. Clinton went out of her way to acknowledge Teneo co-founder Declan Kelly, who worked as a special envoy at the State Department before starting the firm.

“And I, too, will acknowledge and thank our former economic envoy, Declan Kelly, who has done so much to help bring more investors to the region, and I thank you for your contributions, Declan,” Mrs. Clinton declared to applause from the crowd.

“Hi everyone — I know we’ve all be exchanging emails about HRC’s trip to Dublin and Belfast next week so I’m sending you details for the visit and if you have any questions, please let me know,” Ms. Abedin wrote. “Hillary is excited many of you are coming and hopes to see as much of you as possible.”

Later in the week, she followed up: “Hi everyone — wanted to follow up on plans for drinks/dinner on the 6th in Dublin. Thanks to John and Declan, we have a reservation at Restaurant 41 at the Residence Club. … See you Thursday!”

The “John” mentioned is hotel magnate John Fitzpatrick, founder of Fitzpatrick Hotel Group and longtime Clinton supporter.

‘De-loop me’

A month earlier, Ms. Abedin took an entirely different tack, asking to step aside from another matter that had arisen inside the State Department that involved Teneo.

“You should de-loop me from these exchanges,” she wrote, using lowercase letters. “… Now that teneo is involved, I feel I have a conflict so best not to be a part of this. It makes me uncomfortable.”

Friends of Ms. Abedin, who spoke only on the condition of anonymity, declined to speculate why she agreed to work on one matter involving Teneo in Ireland and recused herself from another.

But they said the special government employee arrangement and temporary job at Teneo were designed more to help her transition to private life after an intense three years as a top deputy to Mrs. Clinton, followed by the birth of her first child and the embarrassment of her husband’s scandal.

“Teneo had plenty of connections inside State already with heavy hitters like Doug Band and Declan Kelly, and they didn’t need Huma to make things happen there,” one friend said. “But Huma needed to sort out what she was going to do next with a new child, a new home in New York, a husband who lost his job in Congress and her longtime boss stepping down. And this arrangement helped.”

Ms. Abedin’s email workload seemed to reflect that transition. As deputy chief of staff, she averaged sending 2,193 emails a month through her official State account, but only 1,103 a month after she became a special government employee in 2012, according to an analysis conducted by Sate Department investigators.

Transition phase

Joe Birkenstock, a partner with Sandler Reiff Lamb in Washington, said special government employee status is given frequently to help government employees as they are leaving their jobs.

“Honestly, nothing about this status has made me stop and question it. This is why the SGE status exists, so that employees can continue on in this multiple-role capacity so they can finish up whatever it is on their way out,” Mr. Birkenstock said. “The concern is whether anybody in that position would use it to advance their own personal financial interest, and it doesn’t look like Huma was in a position to do that.”

“Multiple employers often mean competing or overlapping interests,” the senator told The Times. “Managing and preventing conflicts of interest are extremely important. In this case, the lack of transparency and the difficulty in determining when a government official is actually working for the government or a private entity make it difficult to determine whether conflicts of interest were avoided appropriately. That’s why the exemption for special government employees should be limited to cases where there is a truly exceptional government need for specialized knowledge or skills that can’t be found any other way. The goal behind asking these questions is to make sure government programs and laws, as well as government officials, are working as intended to serve the public, not other employers.”

Questions of propriety

The questions about blurred lines are taking on new significance as both the State Department inspector general and Mr. Grassley’s committee investigate whether Ms. Abedin’s special government employee contract was properly executed and accounted for and whether she engaged in any conflicts of interest.

David Bossie, a former congressional investigator and founder of the conservative group Citizens United, which sued to win access to Ms. Abedin’s emails, said the messages “pull back the curtain on how the Clintons do business.”

“The Clintons and their cronies know no shame and will do anything to enrich themselves or their friends and allies,” Mr. Bossie said. “Using the foundation, using a private corporation and using the assets of the State Department all for the enrichment of their friends [are] all part of their normal business.”

In a letter addressed to the State Department about her consulting work, Ms. Abedin insisted she never intervened in State Department matters on behalf of Teneo clients and simply provided “strategic advice and consulting services to the firm’s management team” while helping “organize a major annual firm event.”

“I was not asked, nor did I undertake, any work on Teneo’s behalf before the Department,” Ms. Abedin wrote in the letter dated July 5, 2013, that Mr. Grassley released.

Investigators determined that Ms. Abedin charged more hours to her special government employee contract than was allowed by law, according to documents obtained by The Times. That alone, however, would not be enough to void the special protections against indirect conflicts of interest that the status afforded her, according to Office of Government Ethics rules. But a finding of a direct conflict of interest could lead to some form of action, officials said.

In New York, the Essex House event attracted a high-power guest list including Harvey Weinstein, Eli Broad, Blackstone co-founders Stephen Schwarzman and Peter Peterson, Silicon Valley impresario Sean Parker, Billie Jean King, George E. Pataki, and New York Police Commissioner Raymond Kelly, along with CEOs and top executives from companies such as Dow Chemical, Coca-Cola, BP and Bank of America, according to a report by the New Republic.

The three former heads of state, Mr. Clinton, Mr. Bush and Mr. Blair, were slated to take the stage at 7 p.m. to give their speeches, but they were delayed backstage — by a long sales pitch made by a Teneo co-founder, the magazine reported.

“Instead of introducing his distinguished guests, he launched into a long-winded sales pitch,” the New Republic reports. “Teneo was the next big thing in executive consulting, he informed the audience. He played a promotional video about the firm. He introduced the heads of Teneo’s divisions, describing their resumes and asking each to stand in turn. Meanwhile, the onetime guardians of the [U.S.-British] special relationship were left loitering awkwardly in the wings. ‘It was unnecessarily inappropriate,’ said one guest. ‘It was flagrant.’ “